inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the completion of the space of polynomials \( \mathcal{P} \) with respect to the norm \( \| a_0 + a_1 x + \dots + a_nx^n\| = |a_0| + |a_1| + \dots + |a_n| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ell^1}
For  $a>0$ , denote by  $S(a)$  the area of the part bounded by the parabolas  $y=\frac 12x^2-3a$  and  $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of  $S(a)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8\sqrt{2}}{3}}
Find all pairs of natural numbers \((x, y)\) such that \(5^x + 3^y\) is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
顶点为 $P$ 的圆锥的轴截面是等腰直角三角形， $A$ 是底面圆周上的点， $B$ 是底面圆内的点， $O$ 为底面圆圆心， $A B \perp O B$ ，垂足为 $B, O H \perp P B$ ，垂足为 $H$ ，且 $P A=4, C$ 为 $P A$ 的中点，求当三棱锥 $O-H P C$ 的体积最大时， $O B$ 的长。答案的形式为 \frac{k \sqrt{m}}{n}，请给出k + m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Evaluate the limit of the infinite product involving prime numbers: $$ \prod_{p \in \mathbb{P}} \frac{p^4+1}{p^4-1} $$ where $\mathbb{P}$ denotes the set of all prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{6}}
Find a function \( f(x) \) such that \( f''(x) = f(x) \), \( f(0) = 0 \), and \( f(x) \to -\infty \) as \( x \to -\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^x - e^{-x}}
Find the Maclaurin series expansion of \( y = \frac{1+x^2+x^4}{1-x^2+x^4} \) up to the term in \( x^6 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2x^2 + 2x^4}
Let $n$ be a two-digit integer represented as $n = 10a + b$, where $a$ and $b$ are the tens and units digits respectively. If $n$ is $k$ times the sum of its digits, find the constant $x$ such that the number formed by interchanging the digits of $n$ is $x$ times the sum of its digits. Express your answer as an algebraic expression in terms of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11-k}
A parabola is given by the equation \(x^2 + 44x = y + 88\). Let \(P\) be a variable point on the parabola and \(Q\) be a fixed point on the plane not lying on the parabola. Determine the angle between the tangent at \(P\) and the line segment \(PQ\) when the square of the distance between \(P\) and \(Q\) is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Determine the Euler characteristic of the manifold of fixed rank matrices, \( \mathcal{M}_r \), which is the set of \( n \times m \) matrices over \( \mathbb{R} \) or \( \mathbb{C} \) with rank \( r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The geometric series \(a+ar+ar^2+\ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the sum of all values of $x$ for which the power series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{2n+1}}{2n+1} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find max number $n$ of numbers of three digits such that :
1. Each has digit sum $9$
2. No one contains digit $0$
3. Each $2$ have different unit digits
4. Each $2$ have different decimal digits
5. Each $2$ have different hundreds digits
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n = 5 }
Let \((X_i)_{i=1}^{20}\sim N_6(\mu,\Sigma)\). Determine the distribution of the quadratic form \((X_1-\mu)^T\Sigma^{-1}(X_1-\mu)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2_6}
Given a quadratic inequality in $x$, $ax^2+2x+b>0$, whose solution set is $\{x|x\neq c\}$, find the range of $$\frac {a^{2}+b^{2}+7}{a+c}$$ (where $a+c\neq 0$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -6]\cup[6, +\infty)}
A and B alternately throw a pair of dice until one of them wins. A wins if he rolls a total of 7, and B wins if he rolls a total of 5. A starts the game. What is the probability that one of the dice shows a face of 3 on the last throw of the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{28}}
Question 74, Given $a \geq b \geq c \geq d>0, a^{2}+b^{2}+c^{2}+d^{2}=\frac{(a+b+c+d)^{2}}{3}$, find the maximum value of $\frac{a+c}{b+d}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7+2\sqrt{6}}{5}}
Determine the order of convergence for the error between the exact solution and the cubic Hermite spline approximation, given that the numerical solution used to construct the spline has an order of convergence $p$ and the function $f$ in the initial value problem is only $C^2$ continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(p, 3)}
Decrypt the RSA message given that the public exponent is \( e = 3 \) and the message \( m < N^{1/3} \), where \( c \equiv m^3 \pmod{N} \). Find the plaintext message \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{c}}
Given the function $f(x) = e^x(\sin x - \cos x)$ for $0 \le x \le 2011\pi$, find the sum of all maximum values of the function $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{e^\pi (1 - e^{2010\pi})}{1 - e^{2\pi}}}
Compute the line integral over the smooth Jordan curve \( \gamma \) in \( \mathbb{R}^2 \setminus \{(0,0)\} \) from \((1,0)\) to \((1,0)\), which winds once around the origin in the clockwise direction:
\[ \int_{\gamma}\frac{y}{x^2+y^2}dx-\frac{x}{x^2+y^2}dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Given a circle with radius \(a\) and a diameter \(AB\), chords are drawn perpendicular to \(AB\), intercepting equal arcs along the circumference of the circle. Find the limit of the average length of these chords as the number of chords tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4a}{\pi}}
Find all positive integers \(n\) such that \(n^3 + n + 1\) is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the value of $k$ for which the quadratic form associated with the bilinear form, given by the matrix \( A = \begin{pmatrix} 0 & 0 & k \\ 0 & k & 0 \\ k & 0 & 0 \end{pmatrix} \), has a signature of \((1, 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k < 0}
Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Given $p$: $|x-3|\leqslant 2$ and $q$: $(x-m+1)(x-m-1)\leqslant 0$, determine the range of the real number $m$ if $\neg p$ is a necessary but not sufficient condition for $\neg q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2,4]}
Given a family of functions $f_b(x)$ such that $f_b(0)=b$ and $f_b(x)=2^af_b(x-a)$, find an expression for $f_b(2x)$ in terms of $f_b(x)$. Provide your answer as a function of $f_b(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{(f_b(x))^2}{b}}
已知点集 $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leq\left(\frac{5}{2}\right)^{2}\right.\right\}, B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$, 则点集 $A \cap B$ 中的整点 （即横、纵坐标均为整数的点）的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Lilly, Fiona, Jack, and Emily are working together to clean 3 rooms. For the first room, a quarter of the time is spent cleaning by Lilly and Fiona combined, while Jack spent 1/3 of the time cleaning, and Emily was responsible for the rest of the cleaning. In the second room, Jack and Emily each handle 25% of the cleaning, leaving Lilly and Fiona to split the remaining 50%. In the third room, Emily spends 40% of the time cleaning, Lilly and Jack both contribute 20% each, and Fiona takes care of the remaining 20%. It takes them a total of 12 hours to clean all 3 rooms. How long, in minutes, was Emily cleaning in the second room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Consider a random walk where $X_1, X_2, \ldots$ are independent and identically distributed random variables with a Bernoulli$(\frac{1}{2})$ distribution, meaning $P(X_i = 0) = P(X_i = 1) = \frac{1}{2}$. Define $S_0 = 0$ and $S_n = \sum_{i=1}^n X_i$. Let $\tau_{1000} = \inf\{ n \ge 1 : S_n = 1000 \}$. Find the expected value $\mathbb{E}[\tau_{1000}]$. Provide your answer as a single numerical value or state if it is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
已知$\textit{a}$,$\textit{b}$,$\textit{c}$,$\textit{d}$∈(0,$\textit{+∞}$),且$\textit{S}= frac{a}{a + b + c} + frac{b}{b + c + d} + frac{c}{c + d + a} + frac{d}{a + b + d}, 则S的取值范围是$__________$\textit{.}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2).}
Bill has the expression $1+2+3+\cdots+8.$ He replaces two different addition symbols with multiplication symbols uniformly at random. The value that he obtains on average can be expressed as a common fraction $\\tfrac{m}{n}.$ Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1912}
Two circles (centres $d$ apart) have radii $15,95$. The external tangents to the circles cut at $60$ degrees, find $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
160}
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1, a_2, \ldots, a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots, a_{i_9}$ ($1 \leq i_1 < i_2 < \ldots < i_9 \leq n$) and $b_i \in \{4, 7\}$, $i = 1, 2, \ldots, 9$ such that $b_1 a_{i_1} + b_2 a_{i_2} + \ldots + b_9 a_{i_9}$ is a multiple of 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9
}
In triangle $ABC$, $a$, $b$, and $c$ respectively are the sides opposite to angles $A$, $B$, and $C$. If $\frac{\cos C}{\cos B}=\frac{2a-c}{b}$, then $B=$ _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Consider the ring \( S=\begin{Bmatrix}\begin{pmatrix}a&b\\0&a\end{pmatrix}: a,b\in \mathbb{Z}\end{Bmatrix} \). Determine the number of endomorphisms of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is $DE+EF$? [asy]
pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4);
draw(a--b--c--d--e--f--cycle);
draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a);
draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b);
draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c);
draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d);
draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f);
label("$A$", a, NW);
label("$B$", b, NE);
label("$C$", c, SE);
label("$D$", d, SW);
label("$E$", e, SW);
label("$F$", f, SW);
label("5", (0,6.5), W);
label("8", (4,9), N);
label("9", (8, 4.5), E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
640
}
The numbers $1, 2,. . . , 10000, $ were placed in the cells of a $100 \times  100$ square, each once; in this case, numbers differing by $1$ are written in cells adjacent to each side. After that we calculated distances between the centers of every two cells whose numbers differ by exactly $5000$. Let $S$ be the minimum of these distances What is the largest value $S$ can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ 50\sqrt{2} $
}
Determine all natural numbers \( n \) such that \( w = \frac{2^{n+1}(1+i)}{(1-i)^{3n+5}} \) is a purely real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 2 \pmod{4}}
Determine the maximum dimension of a subspace M of the space of 6x6 skew-symmetric matrices over the real numbers, where all non-zero elements of M have rank 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the smallest constant \( c \) such that for any function \( f \in C^4([0,1]) \) and any \( x, y \in [0,1] \), the following inequality holds:
\[ \left\vert\frac{f(x)-f(y)}{x-y}-\frac{f'(x)+f'(y)}{2}\right\vert\le c\vert x-y\vert ^2\max_{\xi\in[0,1]}\vert f'''(\xi)\vert \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Evaluate the definite integral \( I = \int_{0}^{\infty} (x^2 - 3x + 1) e^{-x} \ln^3(x) \, dx \) using the series expansion \( e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6\gamma}
Let $r_1, r_2, \ldots, r_{20}$ be the roots of the polynomial $x^{20} - 7x^3 + 1$. If \[\frac{1}{r_1^2+1} + \frac{1}{r_2^2+1} + \cdots + \frac{1}{r_{20}^2+1}\] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are positive coprime integers, find $m+n$. \(\text{Express your answer as a single integer.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
When $m=$______, the equation $\frac{5x}{x-2}+1=\frac{m}{x-2}$ will have a repeated root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
If the function $f(x)= \frac{x^{3}}{3}- \frac{a}{2}x^{2}+x+1$ has an extreme point in the interval $\left( \frac{1}{2},3\right)$, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(2, \frac{10}{3}\right)}
Identify a group $G$ that contains elements $a$ and $b$ such that $a^2 = e$, $b^2 = e$, and the order of the element $ab$ is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D_\infty}
X and Y are jointly normal random variables with a common mean of 0, a common variance of 1, and a covariance of 1/2. Calculate the probability that X + Y is less than or equal to \( \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8413}
A survey was sent to some customers, 7 of whom responded. Then the survey was redesigned and sent to another 63 customers, 9 of whom responded. The response rate increased by approximately 5% from the original survey to the redesigned survey. How many customers were sent the original survey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Determine the quotient of \( \mathbb{C}^* \) by the inverse action, where the map \( f: \mathbb{C}^* \longrightarrow \mathbb{C}^* \) is defined as \( z \mapsto z^{-1} \). What is the resulting quotient space?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Let \( c \) be a complex number, and consider the equation that holds for every complex number \( z \):
\[(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)\]
where \( r \), \( s \), and \( t \) are distinct complex numbers. Determine the number of distinct possible values of \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that $\alpha$, $\beta$, and $\gamma$ are the roots of the polynomial $x^3 - x^2 + px - 1 = 0$, and that $(\alpha^3 + 1)(\beta^3 + 1)(\gamma^3 + 1) = 2019$, find the product of all possible values of $p$. Note that $p$ may be a complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
Find the maximal number of red points in a critical coloring of the plane, where each point has coordinates in the set $S=\{0,1,\ldots,99\}$, such that for each $i,j\in S$, at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$, and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is red. The coloring should lose its critical property if any red point is recolored to white.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
In a 1000 by 1000 grid, an s-tile is defined as a path of squares that makes two turns in opposite directions and has a length of at least 4. What is the maximum area that can be covered by s-tiles in this grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
(5) In a simple graph with eight vertices, what is the maximum number of edges in a graph that does not contain a quadrilateral? (A simple graph is one where no vertex is connected to itself, and any two vertices are connected by at most one edge.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Define the sequences $c_n$ and $d_n$ as follows: $c_1 = 6$, $c_{n+1} = (c_n + 1)!$, $d_1 = 6$, $d_{n+1} = 6^{d_n + 1}$. Determine which is larger, $c_{d_2}$ or $d_{c_2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d_{c_2}}
In a language with only three letters - $A, B$, and $C$, a word is defined as a sequence of exactly 100 letters, with exactly 40 consonants and 60 letters being $A$. Determine the maximum number of such words that can be selected such that any two selected words have at least one position where they both have consonants, but different consonants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{40}}
The sequence ${a_n}$ satisfies the equation $a_{n+1} + (-1)^n a_n = 2n - 1$. Determine the sum of the first 80 terms of the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3240}
Evaluate the double integral \( J = \int\limits_0^1 \int\limits_0^1 \frac{x-y}{(x+y)^3} \, dy \, dx .\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given a positive integer \(k\), let \(S_{k}\) be the set of all visible lattice points \((x, y)\) such that \(x^{2}+y^{2}=k^{2}\). A lattice point is visible if the line segment connecting the point and the origin does not pass through another lattice point. Let \(D\) denote the set of all positive divisors of \(2021 \cdot 2025\). Compute the sum \(\sum_{d \in D}|S_{d}|\), where \(|A|\) denotes the number of elements in the set \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a random variable $X$ with the cumulative distribution function (CDF):
\[ F(x) = \begin{cases} 1 - \left(\frac{2000}{2000 + x}\right), & x \ge 0 \\ 0, & \text{otherwise} \end{cases} \]
Let $Y = \max(0, X - 500)$. Calculate the 60th percentile of the positive values of $Y$, ignoring values of $Y$ where $0 \le X \le 500$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3750}
If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{4}}
Evaluate the limit: \[ \lim _{x\to \infty }\frac{\left(x!\right)^3\left(3\left(x+1\right)\right)!}{\left(\left(x+1\right)!\right)^3\left(3x\right)!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
A collection of stained glass windows consists of 9 squares arranged in a 3x3 grid. Each square can be either red or blue, with k of them being red. The set is designed such that any possible window configuration can be achieved by rotating or flipping one unique window from the set. Given that the total number of red squares in the set exceeds 100, determine the value of k. Use Burnside's lemma, considering the 8 isometries of a square: the identity, 3 rotations (90°, 180°, 270°), and 4 reflections (vertical, horizontal, and two diagonals).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If $n \in \mathbb{N}^*$, and $n$ is an odd number, then the remainder obtained when dividing $6^{n}+C_{n}^{1} \cdot 6^{n-1}+C_{n}^{2} \cdot 6^{n-2}+...+C_{n}^{n-1} \cdot 6$ by $8$ is $\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
How many ways can $8$ mutually non-attacking rooks be placed on a $9 \times 9$ chessboard so that all $8$ rooks are on squares of the same color?

(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40320}
Let $a_1, a_2, \ldots, a_{100}$ be positive integers, satisfying
$$
\frac{a_1^2+a_2^2+\ldots+a_{100}^2} {a_1+a_2+\ldots+a_{100}}=100.
$$
What is the maximal value of $a_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{550}
Find the number of ways to color all natural numbers from 1 to 20 in blue and red so that both colors are used and the product of all red numbers is relatively prime to the product of all blue numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
There exists complex numbers  $z=x+yi$  such that the point  $(x, y)$  lies on the ellipse with equation  $\frac{x^2}9+\frac{y^2}{16}=1$ . If  $\frac{z-1-i}{z-i}$  is real, compute  $z$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \frac{3\sqrt{15}}{4} + i}
Determine the angle between the hour and minute hands of a clock showing 1:10, assuming both hands move at constant speeds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25^\circ}
For any positive integer $n$, let $x_n = \mathrm{C}_{2n}^{n}$. Find all positive integers $h$ (where $h > 1$) such that the sequence $\left\{ x_n \right\}$, when taken modulo $h$, becomes a periodic sequence starting from a certain term.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a group of people participating in some curriculum, 25 of them practice yoga, 18 study cooking, 10 study weaving, 4 of them study cooking only, some of them study both the cooking and yoga, 4 of them participate all curriculums. If 5 people study both cooking and weaving, how many people study both cooking and yoga?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}