inclusionAI/Ring-lite-rl-data huggingface dataset:
Integrate the vector field \( F(x,y,z) = \frac{1}{\sqrt{x^2+y^2}}(y,-y,1) \) over the surface of the paraboloid defined by \( z = 1 - x^2 - y^2 \) for \( 0 \leq z \leq 1 \), which corresponds to the region \( x^2 + y^2 \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{3}}
Determine the value of $p$ for which the integral \( \int \frac{dx}{\sqrt{x+2}(x-3)} \) equals \( \frac{2}{\sqrt{3}}\arctan\left(\frac{\sqrt{x+2}}{\sqrt{3}} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \(n\) numbers such that no two of the selected numbers include one number being seven times another. What is the maximum value of \(n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1763}
Calculate the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Determine the number of roots of the polynomial \( f(z) = 1 + 6z^3 + 3z^{10} + z^{11} \) that lie within the annulus defined by \( 1 < |z| < 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the residue of the function \( f(z) = \frac{1}{z^2+1} \) at the point \( z = i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{i}{2}}
Determine the number of roots of the equation \(z^4 + z^3 + 1 = 0\) that lie in the first quadrant of the complex plane using Rouche's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the domain of definition for the function \( g(x) = \int_{0}^{\infty}\frac{t^x}{1+t^2 \cos^2 t}dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0)}
Find a tight upper bound, expressed as an arithmetic function \( f(n, k) \), for the length of the shortest word in the language generated by a context-free grammar \( G = (\Sigma, N, R, S) \), where \(|N| = n\) is the number of non-terminals, \(k\) is the maximum length of a production rule, and \(L(G)\) is non-empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^{n-1}}
Find a closed-form expression for the recursive formula \( f(n) = 1+\frac{1}{n}\sum_{i = 0}^{n - 1} f(i) \) with the base case \( f(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_n}
Evaluate the line integral \( \int_{C_2} \frac{x\vec{e_1}+y\vec{e_2}}{x^2+y^2} d\vec{r} \) where \( C_2 \) is a square with vertices \((1,1), (-1,1), (-1,-1), (1,-1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the greatest integer $a$ such that the inequality \(x^2 - ax - \ln x + e^{\sin x} - 1 > 0\) is satisfied for all \(x > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the residue of \( \frac{1}{(e^z-e)^3} \) at \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e^3}}
A tank with a diameter of $40\,\rm m$ and a height of $15\,\rm m$ is initially $50\%$ full with $92$-Octane fuel. The fuel is being drained at a rate of $500\,\rm m^3$ per hour, while $87$-Octane fuel is being added at a rate of $3000\,\rm m^3$ per hour. Assuming the fuel mixes evenly, determine how many hours it will take for the fuel in the tank to become $91$-Octane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.77}
Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent random variables, where \(X_{2k-1}\) follows a Rademacher distribution with \(P(X_{2k-1}=1) = P(X_{2k-1}=-1) = \frac{1}{2}\), and \(X_{2k}\) follows a Cauchy distribution with probability density function \(f(x) = \frac{1}{\pi(1+x^2)}\). Determine the limit:

\[ \lim_{n \to \infty} P\left(\frac{X_{1} + \cdots + X_{2n}}{\sqrt{n}} \ge 2 \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
What is the minimum number of orthogonal matrices required to express the matrix $M_{i,j}$, which is a standard basis matrix of size $n > 1$, as a linear combination of those orthogonal matrices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the double integral \( \iint_{D} xy \, dA \) where \( D \) is the region bounded by the lines \( y=x \), \( y=x+4 \), \( x=0 \), and \( x=2 \). Use the change of variables \( x=2u \) and \( y=2u+v \). Express your answer as a decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26.6667}
Find the minimum value of the sum \(r_1 + r_2\) for two distinct non-negative integers \(r_1\) and \(r_2\) such that the binomial coefficient \(\binom{1999}{r}\) is even for both \(r_1\) and \(r_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Let $A$ be an $n \times n$ matrix with integer entries, each with an absolute value bounded by a constant $C$. Determine whether the entries of the inverse matrix $A^{-1}$ can grow more rapidly than exponentially with respect to $n$, or if they are bounded by $(C + O(1))^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(C + O(1))^n}
An Abelian group G is generated by elements x and y, where the order of x is 16 and the order of y is 24. Given that x^2 = y^3, determine the order of the group G.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given a connected graph with an independence number $k > 1$, determine the maximum possible value for the path number of the graph in terms of $k$. The independence number of a graph $\mathbf{G}$ is the maximum number of pairwise non-adjacent vertices in $\mathbf{G}$. Express your answer in terms of $k$. \( \text{path number} = f(k) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
Let \((a_1, a_2, \ldots, a_{2014})\) be a permutation of \((1, 2, 3, \ldots, 2014)\). Define \(S_k = a_1 + a_2 + \cdots + a_k\) for \(k = 1, 2, \ldots, 2014\). Find the maximum possible size of the set \(A = \{S_i \mid S_i \equiv 1 \pmod{2}\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1511}
Find the abelianization of the group \( G = \langle a, b \mid ab^2 = b^2a,\ a^4 = b^3 \rangle.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Given that $689\Box\Box\Box20312 \approx 69$ billion (rounded), find the number of ways to fill in the three-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["500"]}
Five diamonds are to be embedded at five equally spaced positions around a circle. If there are three different colors of diamonds available, how many distinct ways can the diamonds be embedded, considering that the positions are indistinguishable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Determine the number of solutions to the inequality \( \frac{p_{n+1} - p_n}{2} \ge \pi(p_n) \), where \( p_n \) and \( p_{n+1} \) are consecutive primes and \( \pi(n) \) is the prime counting function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A park contains two adjacent equilateral triangles formed by five straight paths, each of length \(a\). Additionally, six shorter paths, each of length \(b\), connect the centers of these triangles to their vertices. Determine the shortest route a watering machine must take to travel along all the paths and return to the starting point \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6a + 8b}
Find the smallest prime number \( p \) such that \( p \), \( p+1 \), and \( p-1 \) each have at least three distinct prime factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131}
For a group with prime order $p$, determine the conditions under which the function $f(k) = k^n$ is invertible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \not\equiv 0 \pmod{p}}
Calculate \( \frac{dF(x)}{dx} \bigg|_{x=0} \) for the function \( F(x) = \int_0^2 \sin(x+l)^2\ dl \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(4)}
Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of
$$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-8
}
Solve the inequality \( \frac{\left(\left|x^{2}-2\right|-7\right)(|x+3|-5)}{|x-3|-|x-1|} > 0 \). Find and write the greatest integer that is a solution to the inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the abelianization of the free product of the groups $\mathbb{Z}_4$ and $\mathbb{Z}_5$. Express your answer in terms of known groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_{20}}
Let $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$, $x_7$ be natural numbers, and $x_1 < x_2 < x_3 < \ldots < x_6 < x_7$, also $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
On the side $AC$ of triangle $ABC$, a circle is constructed such that it intersects sides $AB$ and $BC$ at points $D$ and $E$ respectively, and is constructed with $AC$ as its diameter. The angle $\angle EDC$ is equal to $30^\circ$, $AE = \sqrt{3}$, and the area of triangle $DBE$ is related to the area of triangle $ABC$ as \(1:2\). Find the length of segment $BO$, where $O$ is the point of intersection of segments $AE$ and $CD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
We will call a natural number interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6310972}
Ship \( P \) spots ship \( Q \), which is moving in a direction perpendicular to \( PQ \), maintaining its course. Ship \( P \) chases \( Q \), always heading directly towards \( Q \); the speed of both ships at any moment is the same (but can change over time). It is clear without calculations that \( P \) travels along a curved path; if the chase lasts long enough, the trajectory of the pursuing ship and the trajectory of the fleeing ship eventually become nearly identical. What will the distance \( PQ \) be then, if initially it was 10 nautical miles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \text{ nautical miles}}
How many positive three-digit integers with a $7$ in the units place are divisible by $21$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
$在某次投篮测试中，有两种投篮方案.方案甲：先在A点投篮一次，以后都在B点投篮；方案乙：始终在B点投篮.每次投篮之间相互独立.某选手在A点命中的概率为 \frac{3}{4} ，命中一次得3分，没有命中得0分；在B点命中的概率为 \frac{4}{5} ，命中一次得2分，没有命中得0分.用随机变量\xi 表示该选手一次投篮测试的累计得分，如果\xi 的值不低于3分，则认为通过测试并停止投篮，否则继续投篮，但一次测试最多投篮3次.$
$若该选手选择方案甲，求测试结束后所得分\xi的数学期望；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3.05$}
Given $f(x) = \frac{x^2}{2x+1}$, $f_1(x) = f(x)$, $f_n(x) = \overset{\overset{f(\cdots f(x)\cdots)}{}}{n\text{个}f}$, then $f_{10}\left(\frac{1}{2}\right) = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3^{1024} - 1}}
A plane is drawn through the vertex of the base and the midpoints of two lateral edges of a regular triangular pyramid. Find the ratio of the lateral surface area of the pyramid to the area of its base, given that the cutting plane is perpendicular to one of the lateral faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
Consider the set $S$ of the eight points $(x,y)$ in the Cartesian plane satisfying $x,y \in \{-1, 0, 1\}$ and $(x,y) \neq (0,0)$. How many ways are there to draw four segments whose endpoints lie in $S$ such that no two segments intersect, even at endpoints?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the hyperbola \( P : \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \) with left and right foci \( B \) and \( C \), point \( A \) lies on \( P \). \( I \) is the incenter of triangle \( ABC \) and the line \( AI \) passes through the point \( (1,0) \). If \( \overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C} \), then \( x+y \) equals ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
A trapezoid $ABCD$, where side $AB$ is parallel to side $CD$, is divided by its diagonals, which intersect at point $M$, into four parts. Determine its area, given that the triangle $AMD$ has an area of $8 \ \mathrm{cm}^{2}$ and the triangle $DCM$ has an area of $4 \ \mathrm{cm}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36\, \mathrm{cm}^2}
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}
Beatrix is going to place six rooks on a $6 \times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{371}
A natural number $n$ is called a "good number" if the column addition of $n$, $n+1$, and $n+2$ does not produce any carry-over. For example, 32 is a "good number" because $32+33+34$ does not result in a carry-over; however, 23 is not a "good number" because $23+24+25$ does result in a carry-over. The number of "good numbers" less than 1000 is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given a tetrahedron with four faces having areas \( S_{1}, S_{2}, S_{3}, S_{4} \), let \( S \) be the largest of these areas. Define \( \lambda = \frac{\sum_{i=1}^{4} S_{i}}{S} \). Find the range of values that \( \lambda \) must satisfy. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2<\lambda \leq 4}
Calculate the flux of the vector field \(\mathbf{a}=x \mathbf{i}+y \mathbf{j}+\sqrt{x^{2}+y^{2}-1} \mathbf{k}\) through the external side of the one-sheet hyperboloid \(z=\sqrt{x^{2}+y^{2}-1}\), bounded by the planes \(z=0\) and \(z=\sqrt{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{3} \pi}
A sequence \( a_1, a_2, \cdots, a_n, \cdots \) is defined such that \( S(a_i) \) denotes the sum of all digits of \( a_i \). For example, \( S(22) = 2 + 2 = 4 \). Given \( a_1 = 2017 \), \( a_2 = 22 \), and \( a_n = S(a_{n-1}) + S(a_{n-2}) \), find the value of \( a_{2017} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1705}
方程$\sin x = \frac { x } { 100 }$的实根个数有 个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Compute $\sin 12^\circ \sin 36^\circ \sin 72^\circ \sin 84^\circ.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
The diagonals of a quadrilateral \(ABCD\) inscribed in a circle of radius \(R\) intersect at point \(M\). It is known that \(AB = BC = a\) and \(BD = m\).

Find the radius of the circumcircle of triangle \(BCM\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{aR}{m}}
How many roots does the equation \(8x(1 - 2x^2)(8x^4 - 8x^2 + 1) = 1\) have on the interval \([0, 1]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the minimum number of convex pentagons needed to form a convex 2011-gon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}
Simplify the following fraction:

$$
\frac{x y^{2}+2 y z^{2}+y z u+2 x y z+2 x z u+y^{2} z+2 z^{2} u+x y u}{x u^{2}+y z^{2}+y z u+x u z+x y u+u z^{2}+z u^{2}+x y z}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{y+2z}{u+z}}
There is a reservoir that receives a certain amount of water per unit of time, and at the same time, water is released from it. With the current inflow and outflow rates, the water in the reservoir can last for 40 days. Due to recent rainfall at the source, the inflow of water to the reservoir has increased by 20%. If the outflow rate is increased by 10%, the water can still last for 40 days. If the water is released at the original rate, how many days can the water last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50 \text{ days}}
Right triangles $T_1$ and $T_2$, have areas of 1 and 2, respectively. A side of $T_1$ is congruent to a side of $T_2$, and a different side of $T_1$ is congruent to a different side of $T_2$. What is the square of the product of the lengths of the other (third) sides of $T_1$ and $T_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{28}{3}$}
Let $f$ be the function defined on the open unit disk:

$$f(x)=\sum_{n=0}^{+ \infty} x^{n^2}$$

Find the limit of $f(x)$ as $x$ approaches $-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. Find the smallest primest number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
A rectangle of size $1000 \times 1979$ is divided into cells. Into how many parts will it be divided if one diagonal is drawn in it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2978}
The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
What is \( \frac{1}{4} \) more than 32.5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.75}
Find the largest integer $x$ such that the number

$$
4^{27} + 4^{1000} + 4^{x}
$$

is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = 1972}
There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Let \( m=30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) and let \( M \) be the set of its positive divisors which have exactly two prime factors. Determine the minimal integer \( n \) with the following property: for any choice of \( n \) numbers from \( M \), there exist three numbers \( a, b, c \) among them satisfying \( a \cdot b \cdot c = m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given an arithmetic sequence $\{a_n\}$ with a common difference of $\dfrac{\pi}{9}$, the sum of the first 8 terms is $6\pi$. Let $\tan\dfrac{\pi}{9}=k$. Find the sum of the first 7 terms of the sequence $\{\tan a_n\tan a_{n+1}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11-7k^2}{k^2-1}}
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.

[i]Proposed by Evan Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2306}
Given the parabola $x^{2}=4y$ with focus $F$ and point $A(-1,8)$, let point $P$ be a point on the parabola, then the minimum value of $|PA|+|PF|$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
As shown in the figure, the large circle is a 400-meter track, with the track length from \(A\) to \(B\) being 200 meters, and the straight-line distance being 50 meters. A father and his son start running simultaneously from point \(A\) in a counterclockwise direction around the track for long-distance jogging exercise. The son runs around the large circle, while the father runs along the straight line whenever he reaches point \(B\). The father takes 20 seconds to run every 100 meters, and the son takes 19 seconds to run every 100 meters. With this pace, during which lap will the son first meet the father again?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that the quadratic inequality $2kx^2+kx+\frac{1}{2}\geq0$ holds for all real numbers $x$, the range of the real number $k$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,4]}
Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9802}