inclusionAI/Ring-lite-rl-data huggingface dataset:
Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
14
}
Find the norm of the integral operator $A$ in $L_2(0,1)$, where $Ax(t) = \int_0^t x(s) \, ds$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Calculate the limit \( \lim\limits_{r\to 0^+}\dfrac{1}{\pi r^2}\iint\limits_{|(x,y)|\le r} f(x,y) \, dS \) given that \( f(x,y) \) is continuous at the origin and integrable in that region. Assume \( f(x,y) \) is bounded in the neighborhood of the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(0,0)}
Let $p$ be the largest prime less than $2013$ for which\[ N = 20 + p^{p^{p+1}-13} \]is also prime. Find the remainder when $N$ is divided by $10^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1781
}
Given the AR(1) process \( \{X_t\} \) defined by:
\[ X_t = 2.62 - 0.84X_{t-1} + \epsilon_t, \quad \epsilon_t \sim WN(0, 2.27) \]
compute the variance of the average:
\[ \overline{X} = \frac{1}{3}\sum_{t=1}^{3} X_t \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.90}
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, let the left and right foci of the ellipse be $F_1$ and $F_2$, respectively. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse at points $A$ and $B$. If the line $AF_2$ intersects the ellipse at another point $C$, and the area of triangle $\triangle ABC$ is three times the area of triangle $\triangle BCF_2$, determine the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{5}}
Given that point $M$ is a moving point on the parabola $y^2 = 8x$, with $F$ being the focus of the parabola, and point $A$ lies on the circle $C: (x-3)^2 + (y+1)^2 = 1$, find the minimum value of $|AM| + |MF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{10}}
Calculate the limit as \( n \to \infty \) for the expression: \[ \lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{4}}
Find the equation of the orthogonal projection of the space curve defined by \( z = x^y \) onto the x-y plane, given that the curve is symmetrical about the plane \( y = x \) and lies to the right of the line \( y = e \) and above the line \( y = x \), where \( x^y > y^x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^y = y^x}
A box contains 14 balls: 7 black balls numbered from 1 to 7 and 7 white balls numbered from 1 to 7. A person picks 2 balls at a time and repeats the process until the box is empty. Determine the number of ways to draw the balls such that each time two balls are picked, one black and one white, the difference between their numbers is at most one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Find the residue of the function \( f(z) = \frac{\sin z}{(z^2+1)^2} \) at \( z = \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2e}}
Given that $A = (\sqrt{2008}+\sqrt{2009}),$ $B = (-\sqrt{2008}-\sqrt{2009}),$ $C = (\sqrt{2008}-\sqrt{2009}),$ and $D = (\sqrt{2009}-\sqrt{2008}),$ find $ABCD.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the monotonically increasing sequence $\{a_n\}$, $a_1=2$, the inequality $(n+1)a_n \geq na_{2n}$ holds for any $n \in \mathbb{N}^*$.  
Find the range of values for $a_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 4]}
Evaluate the limit: \[ \lim_{n \to +\infty}\left(\frac{1^p + 3^p + \ldots + (2n-1)^p}{(2n)^p} - \frac{n}{p + 1}\right) \] where \( p \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Estimate the number of floating-point operations (flops) required to solve the linear system \(Ax = b\) to ten-digit accuracy using the Conjugate Gradient method, where \(A\) is a dense symmetric positive definite \(1000 \times 1000\) matrix with a condition number \(\kappa(A) = 100\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240000000}
Determine the value of \(a\) such that the matrix \(A\) is not diagonalizable, where \(A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the values of \(\theta\) for which the set \(S = \{x \in \mathbb{R}^n : |x-a| \leq \theta|x-b|\}\) forms a ball.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1)}
Determine the Big-O notation for the function \((n^n + n2^n + 5^n)(n! + 2^n)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^n \cdot n!)}
Calculate the surface area formed by the rotation around the polar axis of the lemniscate $\rho = a \sqrt{\cos 2 \varphi}$ over the interval $0 \leq \varphi \leq \pi / 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi a^2 (2 - \sqrt{2})}
Let $f(x)$ be a differentiable function on $(-\infty, 0)$, with the derivative $f'(x)$, and suppose $2f(x) + xf'(x) > x^2$. Determine the solution set for the inequality $(x+2016)^2f(x+2016) - 9f(-3) > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2019)}
Find the value of the infinite sum:
$$1 + \frac{0.9}{11} +\frac{0.99}{11^2} +\frac{0.999}{11^3}+\cdots$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1189}{1090}\)}
Equilateral triangles $A B F$ and $B C G$ are constructed outside regular pentagon $A B C D E$. Compute $\angle F E G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48^{\circ}}
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $\overrightarrow{m}=(a+c,-b)$, $\overrightarrow{n}=(a-c,b)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=bc$, then $A=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {2\pi}{3}}
A cone with a base radius of 2 and a slant height of \( c \) contains three spheres, each with radius \( r \). The spheres touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}-1}
Suppose $f$ is a function defined on the real numbers $\mathbb{R}$ that is differentiable and not identically zero. If $f$ satisfies the functional equation $xf(x) - yf(y) = (x - y)f(x + y)$ for all $x, y \in \mathbb{R}$, what is the value of $f'(2x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given $0 \leq x \leq \pi$ and $-\frac{1}{2} < a < 0$, find the minimum value of the function $f(x) = \cos^2x - 2a\sin x - 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2a - 1}
Given the complex number $z$ that satisfies the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|x-1-i|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9\sqrt{5}}{10}}
Find all positive integers $x$ and $y$ that satisfy the Diophantine equation: $$12x^4 - 6x^2 + 1 = y^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 13)}
Calculate $\displaystyle \iiint_A(xy+z+1)dxdydz$, where $A$ is the region bounded by the cones $z=2+\sqrt{x^2+y^2}$ and $z=6-\sqrt{x^2+y^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{80}{3}\pi\)}
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Calculate the following limit:
$$\lim_{n\to\infty} \frac{\sum_{m=1}^{\lfloor n/2 \rfloor} \frac {n!}{(n-2m)! 2^m m!}}{\sum_{m=1}^{\lfloor n/4 \rfloor} \frac {n!}{(n-4m)! 2^{2m} m!}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$1991^{2000}$ 除以 $10^{6}$ ，余数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{880001}
已知 $I_{n}=
\left(1+\frac{i}{2}\right)\left(1+\frac{i}{4}\right)\left(1+\frac{i}{16}\right) \cdots\left(1+\frac{i}{2^{2^{n}}}\right)$, 求 $\left|I_{n}\right|$.\n\n原始答案的格式为\sqrt{\frac{m}{n}\left(1-\frac{1}{2^{2^{n+2}}}\right)}, 请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball of radius $10$, as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
In a 20 over cricket match, taking into account no wide balls, no wides, no extras, no overthrows, as well as a maximum of 3 sixes allowed per over, and only two fielders allowed outside the 30-yard circle for the first 10 overs, and five fielders allowed outside for the last 10 overs. No bowler is allowed to bowl more than 4 overs. What is the maximum number of runs that a batsman can score in an ideal case scenario with these additional constraints?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Four square pieces of paper with side lengths of $2, 3, 4,$ and $5$ are placed on a table. If one piece of paper of each side length is removed, the areas covered by the remaining three pieces decrease by $2, 3, 4,$ and $5$ respectively. What is the total area covered by the four pieces of paper?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Find the greatest exponent $k$ for which $2001^k$ divides $2000^{2001^{2002}} + 2002^{2001^{2000}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2001}
In $\triangle ABC$, $\angle C = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 + 12\sqrt{2}}
Sasha has  $10$  cards with numbers  $1, 2, 4, 8,\ldots, 512$ . He writes the number  $0$  on the board and invites Dima to play a game. Dima tells the integer  $0 < p < 10, p$  can vary from round to round. Sasha chooses  $p$  cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1023}
There are 15 boys and 10 girls in a class. If a certain number of students are selected at random, there are 1050 ways that 1 girl and 2 boys are selected. How many students are selected at random?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a + b\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A small group consists of $3$ members. Each member participates in labor for $1$ day within a week (calculated as $7$ days). If the labor dates can be arranged randomly, what is the probability that the $3$ members will participate in labor on different days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{30}{49}}
Suppose $X_1, \ldots, X_n$ are independent and uniformly distributed on the interval $[0, 1]$. What is the expected value of the ratio of the second largest to the largest of these random variables?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)/n}
The distances between the points are given as $A B = 30$, $B C = 80$, $C D = 236$, $D E = 86$, $E A = 40$. What is the distance $E C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
In a $4 \times 4$ grid, numbers are written in such a way that the sum of the numbers in the neighboring cells (cells that share a side) of each number is equal to 1.

Find the sum of all the numbers in the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The three medians of a triangle divide its angles into six smaller angles, among which exactly \( k \) are greater than \( 30^\circ \). What is the maximum possible value of \( k \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find all real functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \[ f(xy+x) + f(y) = f(xy+y) + f(x) \] for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax + b}
The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is 12. Point \( K \) lies on the extension of edge \( BC \) at a distance of 9 from vertex \( C \). Point \( L \) on edge \( AB \) is at a distance of 5 from \( A \). Point \( M \) divides segment \( A_1 C_1 \) in a ratio of 1:3, starting from \( A_1 \). Find the area of the cross-section of the cube by the plane passing through points \( K \), \( L \), and \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
Find the limit of the following Riemann sum as $n$ approaches infinity:

$$\lim _{n\to \infty }\frac{1}{n^3}\sum _{k=1}^{2n}\frac{k^2}{8}\arcsin \left(\frac{k}{2n}\right) = L$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{6} - \frac{2}{9}\)}
Determine the minimum number of keys required to ensure that any 40 out of 50 drivers can operate 40 cars, each with a unique lock.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{440}
Compute the number of integers \(n \in\{1,2, \ldots, 300\}\) such that \(n\) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. The line $AB$ intersects $CD$ at $E$, and the line $DA$ intersects $CB$ at $F$. Given that $OE = 13$ and $OF = 14$, let $\cos \angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime integers. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
In the following figure, the bigger wheel has circumference  $12$ m and the inscribed wheel has circumference  $8 $ m.  $P_{1}$  denotes a point on the bigger wheel and  $P_{2}$  denotes a point on the smaller wheel. Initially  $P_{1}$  and  $P_{2}$  coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the measure of the set of all irrational numbers $x$ in the interval $[0,1]$ such that, in its decimal representation, if a $1$ appears, it is immediately followed by a $2$. Assume the decimal representation does not terminate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\boldsymbol{g}$ be an $N$-by-$1$ complex Gaussian random vector with distribution $CN(0, \boldsymbol{I})$. Consider an $N$-by-$M$ complex matrix $\boldsymbol{A}$, whose columns are orthogonal and have unit norm. Denote the $m$th column vector of $\boldsymbol{A}$ by $\boldsymbol{a}_m$. Determine the distribution of the scalar $x$ as $M$ approaches infinity:

$$x = \frac{1}{M} \sum_{m=1}^{M} \boldsymbol{g}^H \boldsymbol{a}_m$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\in\mathbb{N}^*)$, and the sequence $\{a_{2n-1}\}$ is increasing while $\{a_{2n}\}$ is decreasing, then $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-\\frac{1}{2}"]}
求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为正整数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Given independent random variables, each with a variance not exceeding 4, determine the number of such variables required so that the probability that the deviation of the arithmetic mean from the arithmetic mean of their expectations is no more than 0.25 exceeds 0.99.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6400}
Determine the degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} + \left(\frac{dy}{dx}\right)^{\frac{2}{3}} + y^2 = 0 \) after eliminating the radicals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
Compute the limit:
\[ \lim_{n\to\infty} e^{-itn^{1/4}}\left(1-\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n}}e^{i\frac{t}{n^{1/4}}} \right)^{n} \] where \( t \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-t^2/2}}
5. Garland (1 b. 6-11). A string of 100 lights was hung on a Christmas tree in a row. Then the lights started switching according to the following algorithm: all lights turned on, after a second, every second light turned off, after another second, every third light switched: if it was on, it turned off and vice versa. After another second, every fourth light switched, then every fifth light, and so on. After 100 seconds, everything was over. Find the probability that a randomly chosen light is on after this (the lights do not burn out or break).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1}
Let \(a, b, c\) be positive integers. All the roots of each of the quadratics \(a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c\) are integers. Over all triples \((a, b, c)\), find the triple with the third smallest value of \(a+b+c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,10,24)}
Given a sector with a central angle of $\frac{4}{3}\pi$ and an area of $6\pi$, which represents the lateral surface of a cone, calculate the volume of the cone. Express your answer in terms of $\pi$ and include the appropriate units and precision.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{5}}{3}\pi}
a 100 meter long train crosses a man standing on the platform in 5 sec . what is the speed of the train ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)? 

(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and
$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$
$$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$
$$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$
$$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$

The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5041}
In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula $$4 \times(\# \mathrm{~A} \text { 's })+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \mathrm{Q} \text { 's })+1 \times(\# \mathrm{~J} \text { 's })$$ Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{197}{1820}}
One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1443}
A cylinder is inscribed in a cone, with the height of the cylinder equal to the diameter of the cone's base. The total surface area of the cylinder is equal to the area of the base of the cone. Find the angle between the generatrix (slant height) of the cone and the plane of its base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan \left( \frac{2(4 + \sqrt{6})}{5} \right)}
Given vectors $\overrightarrow {m}=(\lambda+1, 1)$ and $\overrightarrow {n}=(4, -2)$. If $\overrightarrow {m} \parallel \overrightarrow {n}$, then $\lambda= \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Yang has the sequence of integers $1,2, \ldots, 2017$. He makes 2016 swaps in order, where a swap changes the positions of two integers in the sequence. His goal is to end with $2,3, \ldots, 2017,1$. How many different sequences of swaps can Yang do to achieve his goal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017^{2015}}