inclusionAI/Ring-lite-rl-data huggingface dataset:
In the quadrilateral $ABCD$, the points $E$, $F$, and $K$ are midpoints of the sides $AB$, $BC$, and $AD$, respectively. It is known that $KE \perp AB$, $KF \perp BC$, and the angle $\angle ABC = 118^\circ$. Find $\angle ACD$ (in degrees).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Evaluate the integral: $$\int_0^\infty J_0(x) \cdot \text{sinc}(\pi x) \cdot e^{-x} \, \mathrm dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In which acute-angled triangle is the value of the product \(\operatorname{tg} \alpha \cdot \operatorname{tg} \beta \cdot \operatorname{tg} \gamma\) minimized?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{27}}
Determine the values of the parameter $p$ for which the improper integral \( \int_{0}^{\pi/2}\tan^{p}(x) \, dx \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Find the last two digits of the number \( n = 2017^{(2018)^{(2019)^{(\cdots)}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Yesterday, Robert drove from City A to City B. Due to the heavy traffic, he drove at an average speed of 50 miles per hour. Today, he drove back from City B to City A along the same route. Because the traffic was lighter, he drove at an average speed of 60 miles per hour. On his way back, he also spent 20 minutes at a rest stop. If the total time he spent driving (excluding the rest stop) corresponds to an average speed of 55 miles per hour, what is the distance between City A and City B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1100}
Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. The number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. What are the last two digits of the sum ZARAZA + ALMAZ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
For an odd prime $p$, determine the minimum number of distinct primitive roots that must be multiplied together to ensure that the product is not a primitive root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $C$ be the set of real numbers $c$ such that there are exactly two integers n satisfying $2c < n < 3c$. Find the expected value of a number chosen uniformly at random from $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit \(\lim_{{n \to \infty}} \left(n - 1 - 2\left(\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n - 1}{2}\right)}\right)^2\right)\) using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a$ and $b$ are coprime positive integers, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Consider a square defined by \( \{(x,y): 0\le x,y \le 1\} \) divided into \( n^2 \) small squares by the lines \( x = i/n \) and \( y = j/n \). For \( 1\le i \le n \), let \( x_i = i/n \) and define \( d_i = \min_{0\le j\le n} \left| \sqrt{1 - x_i^2 } - j/n\right| \). Determine the value of \( \lim_{n\to\infty} \sum_{i=1}^n d_i \) if it exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
In the rectangular coordinate system, a point (-16, 0) lies on circle C, and the maximum possible value of the radius of C is 16. What is the other point on the circle that lies on the x-axis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(16, 0)}
Let p be the perfect number 496, and q be the perfect number 8128. A positive integer is a perfect number provided that the sum of all its positive factors, including 1 and the number itself, is equal to 2 times the number. What is the sum of the reciprocals of all the positive factors of the perfect numbers p and q?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
在极坐标系中，圆 $\rho=3$ 被直线 $\theta=\frac{\pi}{3}$ 分成两部分的面积之比是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
We will call a natural number interesting if all its digits, except for the first and the last, are less than the arithmetic mean of their two neighboring digits. Find the largest interesting number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96433469}
For every $n \ge 2$, $a_n = \frac{\sqrt[3]{n^3 + n^2 - n - 1}}{n}$. Find the least value of positive integer $k$ satisfying $a_2a_3\cdots a_k > 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Consider the $100 \times 100$ grid of points with integer coordinates $S=\{(x,y) \in \mathbb{Z}^2\mid$ $1 \le x \le 100,$ $1 \le y$ $\le$ $100\}.$ A set $C$ is formed by selecting each $p \in S$ with probability $\tfrac{1}{2}$ uniformly at random. The expansion of $C$ is defined as the set of points $q \in S$ such that $\min_{p \in C} d(q,p) \le 1,$ where $d(q,p)$ denotes the Euclidean distance between $q,p.$ If the expected size of the expansion of $C$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
77407
}
In ten years, I'll be twice my brother's age, three times my sister's age, and four times my youngest brother's age. The sum of our ages will then be 88 years old. My brother is 3 years older than my sister, and my youngest brother is exactly half my sister's age at the moment. If my youngest brother is currently 4 years old, how old am I now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Find the minimum value of the expression:
\[ \sum_{i=1}^n \frac{1}{\prod_{j \ne i} |x_j - x_i|} \]
where \(x_1, x_2, \ldots, x_n\) are real numbers in the interval \([-1, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-2}}
In the tetrahedron \(ABCD\), the ratios of the lengths are:
\[ BD : CD : AB : AC : AD : BC = \sqrt{3} : \sqrt{2} : 1 : 1 : 1 : 1 \]

Find the angle between \(AD\) and \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Determine the number of distinct ways to arrange 3 red, 2 yellow, 5 green, and 2 blue indistinguishable balls around a square table with three balls on each side, considering rotations as identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41580}
Evaluate the line integral \( \int_{C} \frac{y\,dx - (x+1)\,dy}{x^2 + y^2 + 2x + 1} \) where \( C \) is the curve \( |x| + |y| = 4 \) oriented counterclockwise for a full revolution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
求使$n ^ { 2 } + 2 0 0 7 n$为完全平方数的自然数n的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006009}
Let \(\xi\) be a Gaussian vector in \(\mathbb{R}^{d}\) with a unit covariance matrix. Let the matrix \(B=\left(b_{ij}\right)_{i, j=1}^{d}\) be symmetric and positive definite (\(B>0\)), i.e., such that \(z^{\top} B z > 0\) for all non-zero vectors \(z \in \mathbb{R}^{d}\). For which \(d \geq 1\) does \(\mathrm{E}\left(\xi^{\top} B \xi\right)^{-1}\) exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d > 2}
Determine the values of real $x$ for which the series \(\sum_{n=1}^{+\infty} \left (\frac{x^2n^2-2|x|^3n}{1+2xn^2} \right)^{7n}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Find the value of the integral \( \int_{-\pi/2}^{\pi/2} f(x) \, dx \) where \( f(x) \) satisfies the functional equation \( f(x) = \cos(x + f(x)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two positive acute angles \( A \) and \( B \) satisfy the equations \( 3\cos^2 A + 2\cos^2 B = 4 \) and \( \dfrac {3\sin A}{\sin B} = \dfrac {2\cos B}{\cos A} \). What is the value of \( A + 2B \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Let \( A \) be a \( 7 \times 7 \) matrix with rank \( 3 \), and \( a \) be a \( 7 \times 1 \) column vector. Determine the smallest possible rank of the matrix \( A + (aa^T) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the fraction with the smallest denominator in the open interval \(\left(\frac{47}{245}, \frac{34}{177}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{99}}
Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Find the least positive integer   $n$   such that there are at least  $1000$   unordered pairs of diagonals in a regular polygon with  $n$   vertices that intersect at a right angle in the interior of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
A Beaver-number is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of Beaver-numbers differing by exactly $1$ a Beaver-pair. The smaller number in a Beaver-pair is called an MIT Beaver, while the larger number is called a CIT Beaver. Find the positive difference between the largest and smallest CIT Beavers (over all Beaver-pairs).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
79200
}
Find the maximum number of linearly independent homogeneous quadratic forms in 4 variables over \( \mathbb{C} \) such that their intersection has a connected component that is positive dimensional and does not lie on a hyperplane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later).
After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? 
*I. Izmest’ev*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \( \lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x}+n^2x}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which completely coincides with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which completely coincides with this road, and the other side is 12 km. The area of the forest is 45 square km more than the sum of the areas of the fields. Find the total area of the forest and fields in square km.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Let \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \). Define \( f_1(x) = f(x) \), \( f_2(x) = f(f_1(x)) \), \( f_3(x) = f(f_2(x)) \), and generally \( f_{n+1}(x) = f(f_n(x)) \). How many solutions does the equation \( f_n(x) = \frac{1}{2} x \) have in the interval \([0, 1]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Given \( n > 2 \) natural numbers such that there are no three equal numbers among them, and the sum of any two of these numbers is a prime number. What is the largest possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the quadratic inequality about $x$, $ax^{2}+bx+2 > 0$, the solution set is $(- \dfrac{1}{2} , \dfrac{1}{3} )$. Find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-14}
An advertising board is illuminated by 100W bulbs. The life of each bulb follows an exponential distribution with an average life of 600 hours. Using the central limit theorem, determine the minimum number of bulbs required to ensure the board is lit for at least 20,000 hours with a probability of at least 0.9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Calculate the focal distance of the ellipse given by the equation $\frac{x^2}{5} + \frac{y^2}{4} = 1$, using the relationship $c^2 = a^2 - b^2$ where $a^2 = 5$ and $b^2 = 4$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{6x\cdot\sin x-6\cdot\sin\left(x^{2}\right)+x^{4}}{x^{5}\left(e^{x}-1\right)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{20}}
In a cage, there are some chickens and rabbits. When counted from the top, there are 35 heads; when counted from the bottom, there are 94 legs. Chickens have 2 legs and rabbits have 4 legs. Calculate the number of chickens and rabbits in the cage. Express your answer as two integers, the number of chickens and the number of rabbits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23, 12}
$m$  and  $n$  are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:

Starting from one of the hexagons, the Donkey moves  $m$  cells in one of the  $6$  directions, then it turns  $60$  degrees clockwise and after that moves  $n$  cells in this new direction until it reaches it's final cell.

At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?

*Proposed by Shayan Dashmiz*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ m^2 + mn + n^2 }
Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \mathbb{F}_2[x]/(x^3+x^2+x+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Calculate the variance of the stochastic integral \( Y_t = \int^2_0 W_t^2 \, dW_t \), where \( W_t \) is a Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知非负实数$a、b、c、x、y、z$满足$a+b+c=x+y+z=1$。
求$\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$的最大值。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{17}
Given the joint density function \( f_{(x,y)} = \begin{cases} x+y, & \text{if } 0<x<1, 0<y<1 \\ 0, & \text{elsewhere}. \end{cases} \), find the density function of the random variable \( Z = XY \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(1 - z)}
Evaluate the complex integral \( \int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz \) where \( C \) is the contour \( |z|=5 \). Use the Residue Theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i}
Evaluate the lambda calculus expression: \[ (\;(\lambda f.\lambda x. f(f(f(x))))\;(\lambda g.\lambda y.g(g(y))) \;)(\lambda x.x+1)(0) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Some of $100$ towns of a kingdom are connected by roads.It is known that for each two towns $A$ and $B$ connected by a road there is a town $C$ which is not connected by a road with at least one of the towns $A$ and $B$. Determine the maximum possible number of roads in the kingdom.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4900}
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}\] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{k+1}{k} \right)^k}
Find the limit of the function \( y(x) \) as \( x \to \infty \), where \( y(x) \) is defined by the integral equation: \[ y(x) = 7 + \int_0^x \frac{4(y(t))^2}{1+t^2} \, dt. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the maximum value of $a$ such that the differential equation $y' = -\frac{1}{4} + y - \frac{y^2}{a}$ results in an unstable negative population of crabs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the flux of \( \nabla \times G \) exiting from the surface \( S = \{(x,y,z): x^2+y^2+z^2 = 4, x^2+y^2 \leq 2, z>0 \} \), where \( G(x,y,z)=(xy,zx,x^2+y^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}\pi}
Evaluate the limit: \[ \lim_{n\to \infty}\left(\sum_{k=1}^{2n} \frac{1}{\binom{2n}{k}}\right)^n \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Find the derivative \( \frac{dy}{dx} \) if \( y = \frac{1}{1 + x^{n-m} + x^{p-m}} + \frac{1}{1 + x^{m-n} + x^{p-n}} + \frac{1}{1 + x^{m-p} + x^{n-p}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In triangle ABC, angle C is a right angle, and CD is the altitude. Find the radius of the circle inscribed in triangle ABC if the radii of the circles inscribed in triangles ACD and BCD are 6 and 8, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
# Task № 8.3

## Condition:

Donkey Eeyore thinks that his clock is 20 minutes slow, but in reality, it is 15 minutes fast. He agreed to meet Piglet on the meadow at 10:00, but then realized he had more things to do and would leave 30 minutes later than planned. What time will Eeyore actually arrive at the meeting? Write the answer in the format HH:MM.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:55}
Given a map \( f: T^2 \rightarrow T^2 \), where the induced homomorphism on the first homology group \( f^{**}: H_1(T^2) \rightarrow H_1(T^2) \) is given by \( f(x+y) = mx + ny \), calculate the degree of the map \( f \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mn}
Evaluate the integral \( \int\int\int_E z \, dV \), where \( E \) is the region above the paraboloid \( z = x^2 + y^2 \) and below the plane \( z = 6y \). Express the integral in cylindrical coordinates and compute the volume, rounding your answer to the nearest hundredth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1908.52}
Given the sets $A={z|z=i+i^{2}+i^{3}+…+i^{n},n∈N^{*}}$ and $B={z|z=z_{1}⋅z_{2},z_{1}∈A,z_{2}∈A}$, determine the number of elements in set $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the contour integral \( \oint_{\gamma} \frac{d z}{z^{2}+4} \), where \( \gamma = |z-2i|=1 \) is a closed loop counterclockwise, using the Cauchy integral formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Given a hyperbola \( x^{2} - y^{2} = t \) (where \( t > 0 \)), the right focus is \( F \). Any line passing through \( F \) intersects the right branch of the hyperbola at points \( M \) and \( N \). The perpendicular bisector of \( M N \) intersects the \( x \)-axis at point \( P \). When \( t \) is a positive real number (not equal to zero), find the value of \( \frac{|F P|}{|M N|} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx \) using contour integration and the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
# 11.3. (7 points)

The base of the pyramid with vertex $P$ is a quadrilateral $A B C D$, where the sum of angles $A$ and $D$ is five times smaller than the sum of angles $B$ and $C$. Find the angle between the planes of the faces $P A B$ and $P C D$, if both are perpendicular to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
There are several solid and hollow circles arranged in a certain pattern as follows:  
●○●●○●●●○●○●●○●●●…  
Among the first 2001 circles, how many hollow circles are there? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{667}
A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The diameter divides a circle into two parts. In one of these parts, a smaller circle is inscribed so that it touches the larger circle at point \( M \) and the diameter at point \( K \). The ray \( MK \) intersects the larger circle a second time at point \( N \). Find the length of \( MN \), given that the sum of the distances from point \( M \) to the ends of the diameter is 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Let $F$ be a ring with characteristic $p$, and let $f(x) = a_0 + a_1x + \cdots + a_nx^n$ be a polynomial in $F[x]$. Define the derivative of $f(x)$ as $f'(x) = a_1 + 2a_2x + \cdots + na_nx^{n-1}$. Consider the homomorphism of abelian groups $D : F[x] \to F[x]$ given by $D(f(x)) = f'(x)$. Determine the kernel of $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F[x^p]}
Given the function $f(x)=\sin (\omega x+\varphi)$ $(0 < \omega < 3,0 < \varphi < \pi)$, if $x=-\frac{\pi}{4}$ is a zero of the function $f(x)$, and $x=\frac{\pi}{3}$ is an axis of symmetry for the graph of the function $f(x)$, then the value of $\omega$ is \_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{7}}
One hundred students each send messages to 50 different students. What is the least number of pairs of students who send messages to each other? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}