inclusionAI/Ring-lite-rl-data huggingface dataset:
The product of uncommon real roots of the two polynomials \( {x}^{4} + 2{x}^{3} - 8{x}^{2} - {6x} + {15} \) and \( {x}^{3} + 4{x}^{2} - x - {10} \) is ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{

6}
If the community center has 8 cans of soup and 2 loaves of bread, with each can of soup feeding 4 adults or 7 children and each loaf of bread feeding 3 adults or 4 children, and the center needs to feed 24 children, calculate the number of adults that can be fed with the remaining resources.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$as a rational number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{4}}
Let \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) be an even function such that \( f(1)=1 \) and \( f(2017) \neq 1 \). For any \( x, y \in \mathbf{Z} \), the following inequality holds:
$$
2 f(x+y)-f(x)-f(y) \leqslant|f(x)-f(y)|.
$$

Find all possible values of \( f(2018) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If ${a}^{2}=3{b}^{2}+3{c}^{2}-2\sqrt{3}bc\mathrm{sin}A$, then angle $C=$_______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi }{6}}
In triangle \( A B C \), the angle \( \angle B \) is equal to \( \frac{\pi}{3} \). A circle with radius 3 cm is drawn through points \( A \) and \( B \), touching the line \( A C \) at point \( A \). Another circle with radius 4 cm is drawn through points \( B \) and \( C \), touching the line \( A C \) at point \( C \). Find the length of side \( A C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
$P(x)$ is a polynomial of degree at most $6$ such that $P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$, and $P(7)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P(8)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{312}
Determine the values of \(\theta \in \mathbb{R}\) for which the set \(S = \{x \in \mathbb{R}^n : |x-a| \leq \theta|x-b|\}\) is convex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Evaluate the integral: $$ \int\limits_0^{\pi/2} \frac{\cos^3x + \sin^3x}{\sqrt{\sin 2x}}dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4}}
设由 1,2,\cdots,9 的所有排列 X=(x_1,x_2,\cdots,x_9) 构成的集合为 A. 对 X\in A，记 f(X)=x_1+2x_2+\cdots+9x_9，M=\{f(X)\ |\ X\in A\}. 求 |M|.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Determine \[ \left\lfloor \prod_{n=2}^{2022} \frac{2n+2}{2n+1} \right\rfloor, \] given that the answer is relatively prime to 2022.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Calculate the flux of the vector field $\mathbb{v} = \operatorname{curl}\mathbb{u}$, where $\mathbb{u} = (xy, xz^2, x^2y)$, downwards through the paraboloid $z = 1 - x^2 - y^2$ for $x \geq 0$ and $z \geq 0$. What is the value of this flux?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the values of \(a \in [0, \infty)\) for which there exists a continuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that \(f(f(x)) = (x-a)^2\) for all \(x \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the minimum value of the expression \[ \left[\cos ^{2}\left(\dfrac{x }{2}\right) + \,\sqrt{\,{1 - \alpha}\,}\,\sin^{2}\left(\dfrac{x }{2}\right)\right] ^{2} + \dfrac{\alpha }{4}\,\sin^{2}\left(x\right) \] where \( 0 \le \alpha \le 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \alpha}
Let $K = \mathbb{Z}_2[x]/\langle x^4 + x^2 + x \rangle$. Find an element $g \in K$ such that $g^2 = x^3 + x + 1$. Provide your answer as a polynomial in $K$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 + x + 1}
What is the minimum number of nodes in a hexagonal grid required to ensure that among any selection of nodes, there will always be two nodes such that the midpoint of the segment joining them is also a node of the hexagonal grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Rachel has two identical basil plants and an aloe plant. She also has two identical white lamps and two identical red lamps she can put each plant under (she can put more than one plant under a lamp, but each plant is under exactly one lamp). How many ways are there for Rachel to put her plants under her lamps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the $p$-adic valuation of $|a-b|_p$, where $a$ is a primitive $p$-th root of unity in $\overline{\mathbb{Q}_p}$ and $b$ is a root of $X^{p-1}+p$ in $\overline{\mathbb{Q}_p}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
2. 12 friends agree to go out for a meal once a week, each time evenly divided into 3 tables. Each table seats 4 people, until any two people have eaten at the same table at least once. How many weeks are needed at minimum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral \( \int_{0}^{1}\sum_{n=1}^{\infty}\frac{9}{\pi n^{2}}\cos^{3}\left(\frac{2 \pi n x}{3}\right) d x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( k < 1 \)

}
Evaluate the double integral: \[ \int_0^1\int_0^1 \frac {x^2-y^2}{(x^2+y^2)^2} \, dy \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the double integral \( \iint_{[0,1]^2} (1+x^2+y^2)^{-3/2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\frac{1}{4}$ of the candies. Then, he shared $\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike had. Later, Mike’s sister took between one to four candies, leaving Mike with five candies in the end. Calculate the number of candies Mike started with initially.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Evaluate the contour integral \( \int_{|z|=2} \sqrt{z^2-1} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi i}
Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.

Determine the maximum possible number of positive integers among the 2018 integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
Benny goes to the market for buying some apples to be distributed between her 9 kids equally. She takes a certain amount of money with her. The cost of each apple is 4 dollars. She buys enough apples to share 5 apples equally between her 18 kids. How much money did Benny take with her to the market?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
设 n=5，L_n(x)=\df{1}{n!}\cdot\df{\dd^n}{\dd x^n}(x^2-1)^n. 记 S=\displaystyle\int_{-1}^1|L_n(x)|^2\ \dd x，求 [S].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{186}
Define a \textit{subsequence} of a string \( \mathcal{S} \) of letters to be a positive-length string using any number of the letters in \( \mathcal{S} \) in order. For example, a subsequence of \( HARRISON \) is \( ARRON \). Compute the number of subsequences in \( HARRISON \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set
$A=\{n \in \mathbb{N}: P(n)$ is non-zero, square free and $S(n)$ is a proper divisor of $P(n)\}$.
Find the maximum possible number of digits of the numbers in $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
In the plane Cartesian coordinate system, consider the points \((i, j)\) where \(i, j = 1, 2, \cdots, 5\). Calculate the number of triangles with vertices at these points that have a positive area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2148}
In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Vasya has thought of six natural numbers: \( a, b, c, d, e, f \). For one ruble, it is possible to specify any two of them and find out their product. Petya knows that any two of the thought numbers are coprime (i.e., they have no common divisors greater than 1). For what minimum amount can he find out all the thought numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Zhukov G.

A teacher is planning to give the children a problem of the following type. He will inform them that he has thought of a polynomial $P(x)$ of degree 2017 with integer coefficients, the leading coefficient of which is 1. Then he will tell them $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$ and separately inform them of the value of the expression $P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right)$. Based on this data, the children are supposed to find the polynomial that the teacher could have thought of. For what smallest $k$ can the teacher formulate such a problem so that the polynomial found by the children will definitely match the one thought of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Let \( A = \frac{2^2 + 3 \cdot 2 + 1}{3! \cdot 4!} + \frac{3^2 + 3 \cdot 3 + 1}{4! \cdot 5!} + \frac{4^2 + 3 \cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2 + 3 \cdot 10 + 1}{11! \cdot 12!} \). Find the remainder when \( 11! \cdot 12! \cdot A \) is divided by 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $a, b, c$ be real nonzero numbers such that $a + b + c = 12$ and \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = 1. \] Compute the largest possible value of $abc - \left(a + 2b - 3c\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find a lower bound for the integral $\int_{0}^{1}\frac{1}{x^p+1}dx$ for $p<-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2(1-p)}$}
While reading about the history of different towns, Jared learned that Euston had 20 more carriages than Norfolk. Norwich had 100 carriages, and Flying Scotsman had 20 more carriages than Norwich. Additionally, Jared found out about two more towns - Victoria, which had 15 carriages fewer than Euston, and Waterloo, which had twice the number of carriages as Norwich. If Euston had 130 carriages, how many carriages are there in all these towns combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{775}
Given a positive integer \( n \) (\( n \geqslant 6 \)), let set \( A \) consist of sums of any 5 consecutive positive integers not greater than \( n \), and let set \( B \) consist of sums of any 6 consecutive positive integers not greater than \( n \). If the number of elements in the intersection of sets \( A \) and \( B \) is 2016, find the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12106}
When $p = \sum\limits_{k=1}^{6} k \text{ ln }{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{16}$}
已知 (n+1)^{\alpha+1}-n^{\alpha+1}<n^{\alpha}(\alpha+1)<n^{\alpha+1}-(n-1)^{\alpha+1}，-1<\alpha<0. 设 x=\ds{k=4}{10^6}\df{1}{\sqrt[3]{k}}，则 x 的整数部分为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14996}
Consider all prisms whose base is a convex 2015-gon.

What is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
[ Dirichlet's Principle (finite number of points, lines, etc.).]  [ Central angle. Arc length and circumference length ]

On a circle of radius 1, a point $O$ is marked and a notch is made to the right with a compass of radius $l$. From the resulting point $O_{1}$, another notch is made in the same direction with the same radius, and this is done 1968 times. After this, the circle is cut at all 1968 notches, resulting in 1968 arcs. How many different arc lengths can result from this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: $$\lim_{n\rightarrow \infty} n \int_{-n}^{n} \cos^2(\pi x) \left(\frac{1}{(x-3)(x-4)} + \frac{1}{(x+3)(x+4)}\right) dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying
$$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{2}{3} }
Given $f(x) = x^5 + 2x^3 + 3x^2 + x + 1$, calculate the value of $v_3$ when evaluating the value of $f(3)$ using Horner's method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Calculate the dimension of the family of zero-dimensional subschemes of degree 25 in \( \mathbb{A}^5_K \) such that \( V(\mathfrak{m}^3) \subset \Gamma \subset V(\mathfrak{m}^4) \), where \( \mathfrak{m} \) is the maximal ideal of the origin in \( \mathbb{A}_K^5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{124}
Determine whether there exists a natural number $k$ such that for any selection of 180 vertices from a regular 360-gon centered at point $O$, there are two vertices $A$ and $B$ among the selected vertices such that the angle $AOB$ measures $k$ degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
What is the maximum length of a closed, non-self-intersecting polyline that follows the grid lines of a $6 \times 10$ checkered field?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Petya and Vasya play a game. First, Petya thinks of a polynomial \( P(x) \) with integer coefficients. Then, several moves are made. During a move, Vasya pays Petya one ruble and names any integer \( a \) of his choice, which he has not named before, and Petya responds by telling how many integer solutions the equation \( P(x) = a \) has. Vasya wins as soon as Petya states the same number of solutions twice (not necessarily consecutively). What is the minimum number of rubles Vasya needs to guarantee a win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the smallest integer \( n \) such that the inequality \[ \sum_{i=n}^{100} \binom{100}{i} \left(\frac{80}{100}\right)^i \left(\frac{20}{100}\right)^{100-i} \geq 0.9 \] holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Calculate the value of the expression \( \frac{1}{n^2} \sum_{i = 0}^{n} \ln\binom{n}{i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$. Find $100m+n$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Evaluate the limit:

$$\lim_{x\to 0} \frac {1-(\cos 2x)^3(\cos 5x)^5(\cos 7x)^7(\sec 4x)^9(\sec 6x) ^{11}}{x^2}$$

Using a Taylor series expansion or L'Hôpital's Rule, find the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-30}
设 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2a+3b+5c$ 的最大值和最小值之差。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The age difference between Asaf and Alexander's age is half the total number of pencils Asaf has. The sum of their ages is 140. Alexander has 60 more pencils than Asaf, and they have together 220 pencils. How old is Asaf?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Determine the number of flips needed to estimate the probability of heads $p$ of a potentially biased coin with an error margin of $\frac{1}{1000}$ and a confidence level of at least $0.99$. What is the minimum number of coin flips required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1658944}
A particle is located at some point on a line. During the first second, it splits in half and the halves move away from each other in opposite directions by a distance of 1 from the original position. During the next second, the resulting particles again split in half and the halves move away from each other in opposite directions by a distance of 1 from their previous positions. When any two particles collide, they annihilate each other. For example, after two seconds, only two particles remain. How many particles will remain after 129 seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $S$ be the set of all point $(x,y)$ in the coordinate plane such that $0 \le x \le \frac{\pi}{2}$ and $0 \le y \le \frac{\pi}{2}$. What is the area of the subset of $S$ for which \[\sin^2x-\sin x \sin y + \sin^2y \le \frac34?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{\pi^2}{6}$}
Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 2 : 1}
A train of a certain length is running with a speed of 65 km/hr. In 5.4995600351971845 seconds, it passes a man who is running at 7 km/hr in the direction opposite to that in which the train is going. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length \(1\). The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. The length of one of the two congruent sides of one of the isosceles triangles is given in the form \(\frac{\sqrt{k}}{m}\). Please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A rectangle is cut at one corner to form a pentagon with side lengths of 13, 19, 20, 25, and 31. Calculate the area of the resulting pentagon. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{745}
Let $X Y Z$ be a triangle with $\angle X Y Z=40^{\circ}$ and $\angle Y Z X=60^{\circ}$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $X Y Z$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $Y Z$, and let ray $\overrightarrow{X I}$ intersect side $Y Z$ at $B$. Determine the measure of $\angle A I B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{\circ}}
A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x| \leq 1$. Compute the number of distinct valid paths the frog can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Given that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$, if the graph of $y=g(x)$ is symmetric about the line $x=2$ and $g(2) = 4$, find $\sum _{k=1}^{22}f(k)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24}
Express the module $M=\mathbb{Z}^3/N$ as a direct sum of cyclic modules, where $N=\{(x,y,z)\in \mathbb{Z}^3\;|\; 2x+3y-5z=0\} \subset \mathbb{Z}^3$. Provide the decomposition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Express the following summation in matrix form: \[ D_{βν} = \sum^{n}_{µ=1}\sum^{n}_{a=1}A_{µν}B_{αµ}C_{αβ} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D = C^\top B A}
Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$60^\circ$

}
Consider the alternating group $A_5$. What is the probability that the subgroup generated by two randomly chosen elements $a$ and $b$ in $A_5$, denoted by $\langle a, b \rangle$, is a cyclic group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{40}}
Determine the cardinality of the set \( M = \{\mu : \mathcal{B}(\mathbb{R}) \to \mathbb{R} : \mu \text{ is a measure}\} \). What is \(|M|\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Given the power function $f(x)=({m}^{2}-2m-2){x}^{{m}^{2}-2}$ is increasing on $(0,+\infty)$, the value of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}